Millican Colloquium: A metric interpretation of reflexivity for Banach spaces--Pavlos Motakis (TAMU) | Department of Mathematics

Millican Colloquium: A metric interpretation of reflexivity for Banach spaces--Pavlos Motakis (TAMU)

Event Information
Event Location: 
GAB 461 (Refreshments at 3:30 in 472)
Event Date: 
Monday, January 30, 2017 - 4:00pm

A Banach space is an object endowed with both linear as well as metrical structure and most interesting properties of such spaces are defined in a context that involves the interaction of both of these structures. The study of subsets of Banach spaces from a purely metrical perspective has been an immensely fruitful process that has unraveled deep connections of Banach spaces to other fields of research such as theoretical computer science. Looking at this topic from a more abstract point of view we consider the question of what properties of separable Banach spaces can be characterized by a purely metrical statement. More specifically, we consider reflexivity. To achieve such a characterization we define for each ordinal number $\alpha<\omega_1$ two metrics $d_{\infty,\alpha}$ and $d_{1,\alpha}$ on the Schreier family $\mathcal{S}_\alpha$. We show that a separable Banach space $X$ is non-reflexive if and only if for every $\alpha<\omega_1$ there exists a map $\Phi:\mathcal{S}_\alpha\to X$ and two positive constants $c$, $C$ so that for all $A,B\in\mathcal{S}_\alpha$

$$cd_{\infty,\alpha}(A,B)\le \|\Phi(A)-\Phi(B)\|\le C d_{1,\alpha}(A,B).$$

This is related to the notion of metrically characterizing a Banach space property via a family of test spaces. This notion was introduced by M. Ostrovskii and it is based on a metrical characterization of superreflexivity by J. Bourgain.

This is joint work with Thomas Schlumprecht.